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Let $\mathbb{C} \mathbb{P}^1 $ the complex projective space and let consider the tautological bundle over $\mathbb{C} \mathbb{P}^1 $:

In the excerpt below (whole document: here ) the map belonging to tautological bundle of $\mathbb{C} \mathbb{P}^1 $ is defined as the canonical map $s: \mathbb{C}^2 \backslash \{0\} \to \mathbb{C} \mathbb{P}^1 $ which assoziates every $(x_0:x_1)$ in canonical way the fiber $s^{-1}(x_0:x_1) = \mathbb{C}*(x_0,x_1)$, therefore the corresponding one dimensional vector space.

My question is how does this constuction yields $\mathcal{O}(-1)$ as line bundle, therefore a twisted sheaf with empty global sections (because of $-1$)?

Source: enter image description here

user267839
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1 Answers1

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First the map $s$ should from $\Bbb C^2 \backslash \{(0,0) \}$ to $\Bbb CP^1$. And this is not exactly this map which gives $\mathcal O(-1)$ as the fiber gives a punctured line for all $[x_0:x_1] \in \Bbb CP^1$.

Your reference didn't wrote the map precisely, so let's do it : the correct construction is to take the projection $X \to \Bbb CP^1$ where $X = \{(z,l) \in \Bbb C^2 \times \Bbb CP^1 : z \in l \}$. Here, the fiber is canonically a line as you said. This shows that $\mathcal O(-1)$ is a line bundle. For see that indeed it has no sections, the simplest thing is to write down the condition with the cocyles.

Remark : it is also interesting to see that the projection $X \to \Bbb C^2$ is the blow-up of $\Bbb C^2$ at the origin.

Nicolas Hemelsoet
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  • Thank you for the answer. One conclusion isn't clear to me: How do you deduce from the fact that the fibers are lines, that $\mathcal{O}(-1)$ is a line bundle? I don't see the relation between $X$ and $\mathcal{O}(-1)$. – user267839 Nov 03 '17 at 22:25
  • @KarlPeter : this is a bit more precise that this, the fiber over $[x_0:x_1]$ is canonically identified with ${z \in \Bbb C^2 : z \in [x_0:x_1]}$, so the line over $[x_0:x_1]$ (considered as point in $\Bbb CP^1$) is exactly $[x_0:x_1]$ (considered as a line in $\Bbb C^2$) ! This is the definition of $\mathcal O(-1)$. Now, you're right that it's not finished yet, for verifying that it is a line bundle I should check that my bundle is locally trivial. On $U = \Bbb C \subset \Bbb CP^1$ you can write a bundle isomorphism $\mathcal O(-1)_{|U} \to U \times \Bbb C$ : (to be continued) – Nicolas Hemelsoet Nov 03 '17 at 22:30
  • We take $U : {x_1 = 1}$ and the isomorphism $\mathcal O(-1)_{|U} \to U \times \Bbb C$ is given by $([x_0:1], \lambda(x_0,1)) \mapsto (x_0, \lambda)$. – Nicolas Hemelsoet Nov 03 '17 at 22:32
  • Insightful. Is it obviously from this definition that if $\mathcal{O}(-1)$ defined in this way, as you did it in your last post, that it't equivalent to the classical definition of $\mathcal{O}(-1)$, namely as the inverse invertible sheaf of Serre's twisted sheaf $\mathcal{O}(1)$ in $Pic(\mathbb{P}^{-1})$? Or do I mixing the matter? – user267839 Nov 03 '17 at 23:01
  • @KarlPeter : What you said is correct. In fact, usually I like to take as definition of $\mathcal O(1) := \mathcal O(-1)^{\vee}$. Alternatively you can compute the cocyles of both. – Nicolas Hemelsoet Nov 03 '17 at 23:08
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    Yes, it becomes slowly clearer. One puzzle piece is missed. It refers to the equivalence (?) of the definitions of $ \mathcal {O}(1) $. So on the one hand $ \mathcal {O}(1):= \mathcal {O}(-1)^V$ where $ \mathcal {O}(-1)^V$ as you wrote above and the other one as introduced in Liu’s AG. Here I opened a new thread treating this question separately: https://math.stackexchange.com/questions/2504257/equivalent-definitions-of-twisted-sheaf-mathcal-o1 – user267839 Nov 04 '17 at 14:03
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    @KarlPeter : good ! If no one will answer to you I can write you something this afternoon or this evening. – Nicolas Hemelsoet Nov 04 '17 at 14:05